Estimation of Demand Chapter 4 Objective Learn how

Estimation of Demand Chapter 4 • Objective: Learn how to estimate a demand function using regression analysis, and interpret the results • A chief uncertainty for managers -- what will happen to their product. » forecasting, prediction & estimation » need for data: Frank Knight: “If you think you can’t measure something, measure it anyway. ” 2002 South-Western Publishing Slide 1

Sources of information on demand • Consumer Surveys » ask a sample of consumers their attitudes • Consumer Clinics » experimental groups try to emulate a market (Hawthorne effect) • Market Experiments » get demand information by trying different prices • Historical Data » what happened in the past is guide to the future

Plot Historical Data • Look at the relationship of price and quantity over time • Plot it » Is it a demand curve or a supply curve? » Problem -- not held other things equal Price D? or S? 2000 1998 2001 1997 1999 1996 1995 quantity Slide 3

Identification Problem • Q = a + b P can appear upward or downward sloping. • Suppose Supply varies and Demand is FIXED. • All points lie on the Demand curve P S 1 S 2 S 3 Demand quantity Slide 4

Suppose SUPPLY is Fixed P • Let DEMAND shift and supply FIXED. • All Points are on the SUPPLY curve. • We say that the SUPPLY curve is identified. Supply D 3 D 2 D 1 quantity Slide 5

When both Supply and Demand Vary • Often both supply and demand vary. • Equilibrium points are in shaded region. • A regression of Q = a + b P will be neither a demand nor a supply curve. P S 2 S 1 D 2 D 1 quantity Slide 6

Statistical Estimation of the a Demand Function • Steps to take: » Specify the variables -- formulate the demand model, select a Functional Form • linear Q = a + b • P + c • I • double log ln Q = a + b • ln P + c • ln I • quadratic Q = a + b • P + c • I+ d • P 2 » Estimate the parameters - • determine which are statistically significant • try other variables & other functional forms » Develop forecasts from the model

Specifying the Variables • Dependent Variable -- quantity in units, quantity in dollar value (as in sales revenues) • Independent Variables -- variables thought to influence the quantity demanded » Instrumental Variables -- proxy variables for the item wanted which tends to have a relatively high correlation with the desired variable: e. g. , Tastes Time Trend Slide 8

Functional Forms • Linear Q = a + b • P + c • I » The effect of each variable is constant » The effect of each variable is independent of other variables » Price elasticity is: E P = b • P/Q » Income elasticity is: E I = c • I/Q Slide 9

Functional Forms • Multiplicative Q=A • Pb • Ic » The effect of each variable depends on all the other variables and is not constant » It is log linear Ln Q = a + b • Ln P + c • Ln I » the price elasticity is b » the income elasticity is c Slide 10

Simple Linear Regression • Qt = a + b Pt + t OLS -ordinary least squares Q • time subscripts & error term • Find “best fitting” line t = Qt - a - b Pt t 2= [Qt - a - b Pt] 2. • min t 2= [Qt - a - b Pt] 2. • Solution: b = Cov(Q, P)/Var(P) and a = mean(Q) - b • mean(P) _ Q _ P Slide 11

Ordinary Least Squares: Assumptions & Solution Methods • Error term has a mean • Spreadsheets - such as » Excel, Lotus 1 -2 -3, Quatro of zero and a finite Pro, or Joe Spreadsheet variance • Statistical calculators • Dependent variable is • Statistical programs such as random » Minitab • The independent » SAS variables are indeed » SPSS independent » Fore. Profit » Mystat Slide 12

Demand Estimation Case (p. 173) Riders = 785 -2. 14 • Price +. 110 • Pop +. 0015 • Income +. 995 • Parking Predictor Coef Stdev t-ratio Constant 784. 7 396. 3 1. 98 Price -2. 14. 4890 -4. 38 Pop. 1096. 2114. 520 Income. 0015. 03534. 040 Parking. 9947. 5715 1. 74 R-sq = 90. 8% R-sq(adj) = 86. 2% p. 083. 002. 618. 966. 120 Slide 13

2 Coefficients of Determination: R • R-square -- % of variation in Q dependent variable that is explained ^ • Ratio of [Qt -Qt] 2 to _ [Qt - Qt] 2. Q • As more variables are included, R-square rises • Adjusted R-square, however, can decline Qt _ P Slide 14

• RULE: If absolute value of the T-tests estimated t > Critical-t, then • Different samples REJECT Ho. would yield » It’s significant. different • estimated t = (b - 0) / b coefficients • Test the • critical t hypothesis that » Large Samples, critical t 2 coefficient equals • N > 30 zero » Small Samples, critical t is on » Ho: b = 0 Student’s t-table » Ha: b 0 • D. F. = # observations, minus number of independent variables, minus one. • N < 30 Slide 15

Double Log or Log Linear • With the double log form, the coefficients are elasticities • Q = A • P b • I c • Ps d » multiplicative functional form • So: Ln Q = a + b • Ln P + c • Ln I + d • Ln Ps • Transform all variables into natural logs • Called the double log, since logs are on the left and the right hand sides. Ln and Log are used interchangeably. We use only natural logs. Slide 16

Econometric Problems • Simultaneity Problem -- Indentification Problem: » some independent variables may be endogenous • Multicollinearity » independent variables may be highly related • Serial Correlation -- Autocorrelation » error terms may have a pattern • Heteroscedasticity » error terms may have non-constant variance Slide 17

Identification Problem • Problem: » Coefficients are biased • Symptom: » Independent variables are known to be part of a system of equations • Solution: » Use as many independent variables as possible Slide 18

Multicollinearity • Sometimes independent variables aren’t independent. • EXAMPLE: Q =Eggs Q = a + b Pd + c Pg where Pd is for a dozen and Pg is for a gross. PROBLEM • Coefficients are UNBIASED, but t-values are small. • Symptoms of Multicollinearity -- high R-sqr, but low t-values. Q = 22 - 7. 8 Pd -. 9 Pg (1. 2) (1. 45) R-square =. 87 t-values in parentheses • Solutions: » Drop a variable. » Do nothing if forecasting Slide 19

Serial Correlation • Problem: » Coefficients are unbiased » but t-values are unreliable • Symptoms: » look at a scatter of the error terms to see if there is a pattern, or » see if Durbin Watson statistic is far from 2. • Solution: » Find more data » Take first differences of data: Q = a + b • P Slide 20

Scatter of Error Terms Serial Correlation Q P Slide 21

Heteroscedasticity • Problem: » Coefficients are unbiased » t-values are unreliable • Symptoms: » different variances for different sub-samples » scatter of error terms shows increasing or decreasing dispersion • Solution: » Transform data, e. g. , logs » Take averages of each subsample: weighted least squares

Scatter of Error Terms Heteroscedasticity Height alternative log Ht = a + b • AGE 1 2 5 8 AGE Slide 23

Nonlinear Forms Appendix 4 A • Semi-logarithmic transformations. Sometimes taking the logarithm of the dependent variable or an independent variable improves the R 2. Examples are: Ln Y =. 01 +. 05 X • log Y = + ß·X. Y X » Here, Y grows exponentially at rate ß in X; that is, ß percent growth period. • Y = + ß·log X. Here, Y doubles each time X increases by the square of X. Slide 24

Reciprocal Transformations • The relationship between variables may be inverse. Sometimes taking the reciprocal of a variable improves the fit of the regression as in the example: • Y = + ß·(1/X) Y E. g. , Y = 500 + 2 ( 1/X) • shapes can be: » declining slowly • if beta positive » rising slowly X • if beta negative Slide 25

Polynomial Transformations • Quadratic, cubic, and higher degree polynomial relationships are common in business and economics. » Profit and revenue are cubic functions of output. » Average cost is a quadratic function, as it is U-shaped » Total cost is a cubic function, as it is S-shaped • TC = ·Q + ß·Q 2 + ·Q 3 is a cubic total cost function. • If higher order polynomials improve the R-square, then the added complexity may be worth it. Slide 26